Hi Lao Tse,
From what I recall, the 13-movement form was created at the request of enthusiasts and/or the Chinese government so that there would be a form to practice for those that had difficulty memorizing or practicing the traditional form. You will note for instance, that the form contains no kicks or one-legged postures. I don't think it was just the Yang family that was asked to do this, but all the main Tai Chi families. Maybe someone else can confirm or correct this.
I think the 16-movement forms were created so that there would be a form that could be taught within a single academic term, teaching one posture per week. At the Tai Chi symposium on Nashville in 2011, each of the five traditional families taught their 16-movement form to give participants a taste of their flavor of Tai Chi.
At the moment, I do not think that either of these forms has a formal place within the Association curriculum, however, they are still taught from time to time and are resource for those who find them useful. Some teachers also use these forms as introductions to Tai Chi and teach them before the traditional forms. I personally used to use them when I had little time to practice, but wanted the psychological satisfaction of doing a complete form.
I hope this helps to answer your question.